Question: Divide the following complex numbers. $\dfrac{2-8i}{3+5i}$
Answer: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate, which is ${3-5i}$. $ \dfrac{2-8i}{3+5i} = \dfrac{2-8i}{3+5i} \cdot \dfrac{{3-5i}}{{3-5i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$. $ = \dfrac{(2-8i) \cdot (3-5i)} {3^2 - (5i)^2} $ Evaluate the squares in the denominator and subtract them. $ = \dfrac{(2-8i) \cdot (3-5i)} {(3)^2 - (5i)^2} $ $ = \dfrac{(2-8i) \cdot (3-5i)} {9 + 25} $ $ = \dfrac{(2-8i) \cdot (3-5i)} {34} $ The denominator now doesn't contain any imaginary unit multiples, so it is a real number. Note that when a complex number, $a + bi$ is multiplied by its conjugate, the product is always $a^2 + b^2$. Now, we can multiply out the two factors in the numerator. $ \dfrac{({2-8i}) \cdot ({3-5i})} {34} $ $ = \dfrac{{2} \cdot {3} + {-8} \cdot {3 i} + {2} \cdot {-5 i} + {-8} \cdot {-5 i^2}} {34} $ $ = \dfrac{6 - 24i - 10i + 40 i^2} {34} $ Finally, simplify the fraction. $ \dfrac{6 - 24i - 10i - 40} {34} = \dfrac{-34 - 34i} {34} = -1-i $